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In this paper we study the sum of powers in the Gaussian integers$\mathbf{G}_k(n):=\sum_{a,b \in [1,n]} (a+b i)^k$. We give an explicit formulafor $\mathbf{G}_k(n) \pmod n $ in terms of the prime numbers $p \equiv 3 \pmod4$ with $p \mid \mid n$ and $p-1 \mid k$, similar to the well known one due tovon Staudt for $\sum_{i=1}^n i^k \pmod n$. We apply this formula to study theset of integers $n$ which divide $\mathbf{G}_n(n)$ and compute its asymptoticdensity with six exact digits: $0.971000\ldots$.

A von Staudt-type formula for $\displaystyle{\sum_{z\in\mathbb{Z}_n[i]} z^k }$